Lesson 4: The Manometer and Applications of Pressure

4.1 The manometer

• A manometer measures the difference between the pressure of a gas in a container and the atmospheric pressure.
• It is a U-shaped tube containing a liquid of known density (usually mercury, or water/oil for small pressures).
• One arm is connected to the gas container; the other is open to the air.

• Gas pressure greater than atmospheric:

  • the liquid is lower in the arm connected to the gas.
  • Using two points at the same level in the same liquid:

  P = P_a + ρ g h

• Gas pressure less than atmospheric: the liquid in the open arm is lower:

  P = P_a − ρ g h

The pressure difference (gauge pressure) is ΔP = P − P_a = ρ g h (positive or negative).

Choice of manometer liquid.

• Mercury for large pressures (dense, so the height difference is small); a light liquid (water, oil) for small pressure differences, which gives a large and readable height difference.
• The liquid must not react with or dissolve the gas and should have a low vapour pressure.

4.2 Worked examples

Example 1. A mercury manometer connected to a gas container shows the mercury in the open arm 24 cm higher than in the arm connected to the container. The gas pressure P = 76 + 24 = 100 cm Hg = 100/76 = 1.316 atm = 1.316 × 1.013 × 10⁵ = 1.333 × 10⁵ N/m².

Example 2. If the mercury in the open arm were 18 cm lower, then P = 76 − 18 = 58 cm Hg (below atmospheric: a partial vacuum).

Example 3 (water manometer). A water manometer shows a height difference of 40 cm. ΔP = ρgh = 1000 × 9.8 × 0.4 = 3920 N/m².

Example 4 (mercury, SI). In a manometer h = 25 cm of mercury, ρ = 13 600, g = 9.8: ΔP = 13 600 × 9.8 × 0.25 = 3.332 × 10⁴ N/m²; the absolute gas pressure = 1.013 × 10⁵ + 0.333 × 10⁵ = 1.346 × 10⁵ N/m².

4.3 Applications of pressure

• 1.
• Blood pressure. Blood is a viscous fluid pumped by the heart through arteries and veins.
• Blood flow is normally steady.
• If it becomes turbulent it produces noise, detected by a stethoscope on the artery while measuring blood pressure.
• Two values are recorded:

• Systolic pressure: the maximum pressure in the artery when the heart muscle contracts and pushes blood from the left ventricle into the aorta. Normal: about 120 torr.
• Diastolic pressure: the minimum pressure when the heart muscle relaxes. Normal: about 80 torr. (The cuff is placed on the upper arm at heart level so that the height of the blood column does not add or subtract ρgh to the reading. Long sitting increases the risk of leg blood clots because blood flow in the legs is slow; moving and exercising the legs helps.)

• 2.
• Car-tyre pressure. A tyre filled with air at high pressure has a small contact area with the road.
• A tyre with too low a pressure has a large contact area, so more friction and more heat, and wear.
• Pressure is measured by a tyre-pressure gauge (a piston pushed against a spring with a graduated scale).

4.4 More worked examples

Example 1 (tyre). A tyre gauge shows 2.5 × 10⁵ N/m² above the atmosphere. Absolute pressure = 2.5 × 10⁵ + 1.013 × 10⁵ = 3.51 × 10⁵ N/m² (3.5 atm).

Example 2. Blood pressure of 120 torr = 120 mm Hg = 0.12 × 13 600 × 9.8 = 1.6 × 10⁴ N/m².